On k-neighbour Packings and One-sided Hadwiger Configurations

We show that in d-dimensional Euclidean space the maximum number of non-overlapping translates of a d-dimensional convex body K that can touch K and can lie in a closed supporting half-space of K is always at most 2 ·3d−1−1, with this bound to be reached only if K is an affine image of a d-cube. Such onesided Hadwiger configurations occur at the boundary of finite packings or near the holes in packings of density 0, so this implies that in d-dimensional Euclidean space any k-neighbour packing by translates of a d-dimensional convex body has positive density for all k ≥ 2·3d−1; and there is a (2·3d−1−1)+-neighbour packing by translates of a d-cube that has density 0. (A packing is called a k-neighbour packing if each packing element has at least k neighbours.) This answers an old question of L. Fejes Tóth (1973).